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12.4 The scaling equation

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This module covers scaling equations and coefficients for the discrete wavelet transform.

Consider the level-1 subspace and its orthonormal basis:

V 1 span φ 1 , n t n
φ 1 , n t 1 2 φ 1 2 t n
Since V 1 V 0 ( i.e. , V 0 is more detailed than V 1 ) and since φ 1 , 0 t V 0 , there must exist coefficients h n n such that
φ 1 , 0 t n ∞ ∞ h n φ 0 , n t
⇔ 1 2 φ 1 2 t n ∞ ∞ h n φ t n

Scaling equation

φ t 2 n ∞ ∞ h n φ 2 t n
To be a valid scaling function, φ t must obey the scaling equation for some coefficientset h n .
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Source:  OpenStax, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5
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